Enjoy the lazy days of summer and some engaging mathematics in the latest issue of the *Monthly*.

Peter Winkler explores the probabilistic and philosophical conundrums facing Sleeping Beauty and those observing her as she is awakened once or twice during her slumber. Arseniy Akopyan and Vladislav Vysotsky study the relation between the length of a curve that passes through a fixed number of points on the boundary of a convex shape in the plane and the perimeter of the shape. Moa Apagodu and Doron Zeilberger use the “freshman’s dream identity” (*a* + *b*)^{p} ≡_{p} *a*^{p} + *b*^{p} to elegantly establish a wide variety of combinatorial results. Philip West and Brian Sittinger stroll among the Gaussian and Eisenstein integers in the complex plane, noting the moats of prime-free regions. And John Bowers and Philip Bowers embed metric spaces isometrically in Euclidean spaces.

In the Notes section, you can count walks and graph homomorphisms, identify the first *k*-Ramanujan prime, see how to develop an infinite product formula for the square root of *n*, learn about a condition under which no basis of a *k*-algebra (*k* a field) is closed under multiplication, and learn about a unexpected connection between Kepler’s equation relating the mean and eccentric anomalies of a planet and the witch of Agnesi.

There are, as ever, problems to challenge you. Finally, Calvin Jongsma meditates on the history of mathematics and its long-standing emphasis on Western culture in his review of *Sourcebook in the Mathematics of Medieval Europe and North Africa*, edited by Victor Katz, Menso Folerts, Barnabas Hughes, Roi Wagner, and J. Lennart Berggren.

— *Susan Jane Colley, Editor*

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## Table of Contents

### The Sleeping Beauty Controversy

p. 579.

Peter Winkler

In 2000, Adam Elga posed the following problem:

Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

This may seem like a simple question about conditional probability, but 100 or so articles(including thousands of pages in major philosophy journals) have been devoted to it. Herein is an attempt to summarize the main arguments and to determine what, if anything, has been learned.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.579

### On the Lengths of Curves Passing Through Boundary Points of a Planar Convex Shape

p. 588.

Arseniy Akopyan and Vladislav Vysotsky

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape *K*, there exist four points on the boundary of *K* such that the length of any curve passing through these points is at least half of the perimeter of *K*. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of *K*. Moreover, the factor ^{1}/_{2} cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.588

### Using the “Freshman’s Dream” to Prove Combinatorial Congruences

p. 597.

Moa Apagodu and Doron Zeilberger

Recently, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their approach and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact, we even combine these two generalizations.We conclude with some super-challenges.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.597

### A Further Stroll into the Eisenstein Primes

p. 609.

Philip P. West and Brian D. Sittinger

If one imagines the Eisenstein primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture.

If the frog’s journey terminates for a given hop size, it implies that a prime-free “moat” greater than the hop size completely surrounds the origin.

In the earlier *Monthly* article “A Stroll Through the Gaussian Primes,” Ellen Gethner, Stan Wagon, and Brian Wick explored this problem in the Gaussian primes and by computational methods proved the existence of a √26-moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist.

In their concluding remarks, Gethner et al. note that “Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1.”

This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. We prove that prime-free neighborhoods of arbitrary radius *k* surrounding an imaginary quadratic prime exist by directly generalizing Gethner’s proof regarding prime-free neighborhoods surrounding a Gaussian prime. By computational means we establish the existence of a √16-moat in the Eisenstein primes and provide an estimate for the bounds of any *k*-moat. A subtle difference between the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.609

### A Menger Redux: Embedding Metric Spaces Isometrically in Euclidean Space

p. 621.

John C. Bowers and Philip L. Bowers

We present geometric proofs of Menger’s results on isometrically embedding metric spaces in Euclidean space.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.621

## Notes

### Counting Walks and Graph Homomorphisms via Markov Chains and Importance Sampling

p. 637.

David A. Levin and Yuval Peres

Hoffman proved a matrix inequality that yields a useful upper bound on the number of walks in a graph. Sidorenko extended the bound on the number of walks to a bound on the number of homomorphisms from a tree to a graph. In this expository note, we give a short probabilistic proof of both results, using the basic identity of importance sampling.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.637

### On the First *k*-Ramanujan Prime

p. 642.

Christian Axler and Thomas Leßmann

In this paper we compute the explicit values for the first *k*-Ramanujan prime for every *k* ≥ 1.000040690557321 by using an elegant characterization of the first *k*-Ramanujan prime, which is established in this paper, and a recent result concerning the existence of prime numbers in short intervals.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.642

### An Infinite Product for the Square Root of an Integer

p. 647.

Amrik Singh Nimbran

This paper contains a presumably new representation for the square root of an integer as an infinite product.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.647

### On Bases That Are Closed Under Multiplication

p. 651.

Tomasz Kania

It is well known that there is no basis for the field of real numbers regarded as a vector space over any proper subfield that is closed under multiplication. Mabry has extended this result to bases of arbitrary proper field extensions. The aim of this short communication is to notice that the proof of the result concerning the reals may be adjusted to a larger class of algebras (including full matrix algebras); thereby, we subsume Mabry’s result.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.651

### Maria Gaetana Agnesi Meets Johannes Kepler

p. 654.

Óscar Ciaurri

We provide a surprising result relating the witch of Agnesi and Kepler’s equation.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.654

## Problems and Solutions

p. 659.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.7.659

## Book Review

p. 667.

*Sourcebook in the Mathematics of Medieval Europe and North Africa* by Victor J. Katz, Menso Folkerts, Barnabas Hughes, Roi Wagner, and J. Lennart Berggren, Eds.

Reviewed by Calvin Jongsma

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.6.667

## MathBits

### The Paul R. Halmos–Lester R. Ford Awards for 2016

p. 636.

### The Other Moore Method

p. 641.

### 100 Years Ago This Month in *The American Mathematical Monthly*

p. 653.

### Another Proof of the Binomial Theorem

p. 658